Linearizing Nonlinear Models

What Is Linearization?

Linearization is a linear approximation
of a nonlinear system that is valid in a small region around an operating
point.

For example, suppose that the nonlinear function is y=x2. Linearizing this nonlinear
function about the operating point x =
1, y =
1 results in a linear function y=2x−1.

Near the operating point, y=2x−1 is
a good approximation to y=x2. Away from the
operating point, the approximation is poor.

The next figure shows a possible region of good approximation
for the linearization of y=x2.
The actual region of validity depends on the nonlinear model.

Extending the concept of linearization to dynamic systems, you
can write continuous-time nonlinear differential equations in this
form:

x˙(t)=f(x(t),u(t),t)y(t)=g(x(t),u(t),t).

In these equations, x(t)
represents the system states, u(t)
represents the inputs to the system, and y(t)
represents the outputs of the system.

A linearized model of this system is valid in a small region
around the operating point t=t0, x(t0)=x0, u(t0)=u0,
and y(t0)=g(x0,u0,t0)=y0.

To represent the linearized model, define new variables centered
about the operating point:

δx(t)=x(t)−x0δu(t)=u(t)−u0δy(t)=y(t)−y0

The linearized model in terms of δx,
δu, and δy is
valid when the values of these variables are small:

δx˙(t)=Aδx(t)+Bδu(t)δy(t)=Cδx(t)+Dδu(t)

Applications of Linearization

Linearization is useful in model analysis and control design
applications.

Exact linearization of the specified nonlinear Simulink® model
produces linear state-space, transfer-function, or zero-pole-gain
equations that you can use to:

Plot the Bode response of the Simulink model.

Evaluate loop stability margins by computing open-loop
response.

Analyze and compare plant response near different
operating points.

Design linear controller

Classical control system analysis and design methodologies
require linear, time-invariant models. Simulink Control Design™ automatically
linearizes the plant when you tune your compensator. See Choosing a Control Design Approach.

Analyze closed-loop stability.

Measure the size of resonances in frequency response
by computing closed-loop linear model for control system.

Linearization in Simulink Control Design

You can use Simulink Control Design to linearize continuous-time,
discrete-time, or multirate Simulink models. The resulting linear
time-invariant model is in state-space form.

Simulink Control Design uses a block-by-block approach
to linearize models, instead of using full-model perturbation.
This block-by-block approach individually linearizes each block in
your Simulink model and combines the results to produce the linearization
of the specified system.

The block-by-block linearization approach has several advantages
to full-model numerical perturbation:

Most Simulink blocks have preprogrammed linearization
that provides Simulink Control Design an exact linearization of
each block at the operating point.

You can configure blocks to use custom linearizations
without affecting your model simulation.

Ability to specify linearization to be uncertain (requires Robust Control Toolbox™)

Model Requirements for Exact Linearization

Exact linearization supports most Simulink blocks.

However, Simulink blocks with strong discontinuities or
event-based dynamics linearize (correctly) to zero or large (infinite)
gain. Sources of event-based or discontinuous behavior exist in models
that have Simulink Control Design requires special handling of
models that include:

Blocks from Discontinuities library

Stateflow® charts

Triggered subsystems

Pulse width modulation (PWM) signals

For most applications, the states in your Simulink model
should be at steady state. Otherwise, your linear model is only valid
over a small time interval.

Operating Point Impact on Linearization

Choosing the right operating point for linearization is critical
for obtaining an accurate linear model. The linear model is an approximation
of the nonlinear model that is valid only near the operating point
at which you linearize the model.

Although you specify which Simulink blocks to linearize,
all blocks in the model affect the operating point.

A nonlinear model can have two very different linear approximations
when you linearize about different operating points.

The linearization result for this model is shown next, with
the initial condition for the integration x0 =
0.

This table summarizes the different linearization results for
two different operating points.

Operating Point

Linearization Result

Initial Condition = 5, State x1 = 5

30/s

Initial Condition = 0, State x1 = 0

0

You can linearize your Simulink model at three different
types of operating points: